Prime-Wall Sturmian Renormalization and Moving-Root Projective Criteria for Wall--Riesz Zero Tomography

We study the growing-height continuation problem for the Wall–Riesz observation model developed in the fixed-band predecessor. The arithmetic prime-wall geometry admits an unconditional reduction to a critical Sturmian two-channel Dirichlet-to-Neumann cocycle. A proof-carrying prefix envelope then gives $$2.919396 < \|B_{\text{crit}}\| < 2.91939629, \qquad T_{\min} = 514.$$ For the induced return dynamics we derive a completely positive Schur derivative cocycle and a source-owned determinant-area resource. On the regular safe skeleton a projectively gauged reciprocal-shear Lyapunov quantity expands uniformly and, from one fixed certified deep anchor, yields a conditional positive-power derivative law. The moving-root unit-run loss is then reduced exactly to a logarithmic budget. Using the actual M-matrix tangent cone together with determinant-area genealogy, we cancel absolute derivative scale and obtain the projective bound $$\frac{p'}{\tau'} < \frac{1}{2} e^{\Omega(P \mid T)}.$$ This gives a sharp sufficient criterion for branch persistence in terms of a pure projective-shape budget. Finally, an exact positive two-channel witness shows that generic completely positive structure does not make the current scalar shape observable autonomous under continuation. Thus the remaining unconditional gate is not numerical scale but lossless ownership of projective information across moving roots.